Oddly, Even the Odd Numbers Are Interesting

Odds are high that you’ve not given much thought to numbers, I’d wager.

You encounter them everyday and you use them, knowing the purposes they serve in your situation.

Beyond that, perhaps you have to solve problems with them. Like if you’re a parent helping your kid with Maths sums.

Still, they may just be numbers to you. A bunch of digits, or simply one, which can convey size and quantity; represent monetary value; label things (e.g. house numbers, car licence plates and product serial numbers), etc.

Now, aren’t they convenient and a boon to all of us?

I’d like to take you beyond that and look at the beauty in the numbers…

A word from Mark:“I love puns, so I didn’t let up on the opportunity for word play with the words ‘odd’ and ‘even’. Plus others too, as you’d soon discover. If you find any part of this article confusing, re-read it (in a punny mood, perhaps) and it should all clear up.”

What Are Odd Numbers?

When shown the number sequence 1, 3, 5, 7, 9, …, we’d recognize these numbers are odd numbers. That’s a basic fact we learnt in school.

But to really understand odd numbers, we need to look at even numbers.

An even number is simply an integer* multiple of 2; like 2, 4, 10, 28, etc. When we list consecutive numbers, even numbers will have other numbers between them: these are called the odds.

So, odd numbers are neighbours to even numbers and vice versa. And every integer is either odd or even — this is referred to as parity.

From this, we know that an odd number is simply one more than the even number preceding it (when looking at numbers in ascending order). Or one less than the even number after it.

Expressed as a formula, an odd number is:

2n + 1

where n is any positive integer* or zero.

In case you are wondering, that 2n part signifies a multiple of two, i.e. an even number.

So, when n = 0, we get 1 as the answer — this is the first odd number.

When n = 1, we have 3 or the second odd. And when n = 4, we get 9, which is the fifth odd number. And so on.

*Note: To simplify the discussion, I’ve stayed with natural numbers# only. Indeed, n can also be a negative integer; so -3 and -11 are also odd while -6 and -24 are even.

Rearranging, we’d get 2n + 1 – 2, which is both familiar and logical. Familiar because 2n + 1 is the definitive odd number; and logical, as consecutive odds always differ by 2.

With this, we get this neat trick below…

To get the n-th odd number, use:

2n – 1

Odd Number Rules

From simple observation, we have the following rules when operating with numbers:

odd + odd = even

odd – odd = even

odd X odd = odd

Again for simplicity, I’ve omitted the more complicated division operation.

Addition and Subtraction are Simple

The first 2 rules aren’t difficult to understand if we perceive odd numbers as offset by one position from their even siblings. Pictorially, we have the following:

So, if the first addend (for Rule 1) or the minuend (for Rule 2) is an odd number, we are starting with an offset position.

And if we depict adding on or taking away an odd number as a “shifting” operation by one position — left or right is immaterial here — we are back to a non-offset position.

In other words, we’ve just toggled the parity, from odd back to even. Hence the first two rules are what they are, naturally.

Multiplication Explained

Multiplication is a little harder to see, but if we think about it as repetitive addition it becomes obvious. Multiplying by 2 means adding a copy of a number to itself — once. By 3 means adding on two copies. And so forth.

So when we multiply an odd number by another odd, we add on an even number of copies of either number. Returning to our picture model, adding an even number means nothing gets shifted; so the result is an odd number, intuitively.

These Aren’t Odd, Are They?

Odd numbers aren’t queer at all. Really.

Consider these mathematical facts…

Distribution of Whole Numbers

In the strict sense, odd numbers make up half** of all whole numbers out there; the rest belong to the evens. If they didn’t, then these numbers would have been few and far between. Like prime numbers, for example.

**Ok, to be really correct, it’s ever so close to 50% as the “odd number” (read “unusual”) zero is considered even. However, if we stretch out to infinity, then odd/even (dis)parity pretty much evens out. See?

Prime Numbers

Every prime number is an odd number except for the first, which is the number 2.

By its very definition, a prime number does not have other factors except for 1 and itself; thus, it’s not hard to understand that multiples of other numbers can’t be prime.

On that same token, even numbers — remember, these are actually multiples of 2 — aren’t prime too except for one odd man out. Which explains clearly the fact about prime numbers.

Consecutive Squares and Their Differences

Now, let’s take it one step further and delve into squares. Not just any square, but squares of consecutive numbers.

Did you know that the difference between the squares of two consecutive numbers is always odd?

More interestingly, did you know that these differences would form the odd number series when all the whole numbers are involved? My boy “discovered” this and I thought it’ll be a fun topic to write about.

Let’s dive in…

Consecutive numbers differ by 1. For two consecutive numbers, we can denote the smaller number by a; its neighbour would then be a + 1.

And we’d write their squares as a2 and (a + 1)2 respectively.

Using a little algebra — parents, I hope you aren’t rusty here — we can expand the larger square of (a + 1)2 into a2 + 2a + 1.

Then, the difference between the two consecutive squares (I’ll call this DoCS for short) becomes:

a2 + 2a + 1 – a2 = 2a + 1

Recognize the result as the formula for an odd number? We’ve just verified the first fact that DoCS is always odd!